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Cluster exchange groupoids and framed quadratic differentials

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Abstract

We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case that arises from an (unpunctured) marked surface, where the exchange graph is modelled on the graph of triangulations of the marked surface, we show that the universal cover of this groupoid can be constructed using the covering graph of triangulations of the surface with extra decorations. This covering graph is a skeleton for a space of suitably framed quadratic differentials on the surface, which in turn models the space of Bridgeland stability conditions for the 3-Calabi–Yau category associated to the marked surface. By showing that the relations in the covering groupoid are homotopically trivial when interpreted as loops in the space of stability conditions, we show that this space is simply connected.

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Acknowledgements

QY would like to thank Aslak Buan and Yu Zhou for collaborating on the prequels to this paper. He is also grateful to Tom Bridgeland and Ivan Smith for inspiring discussions. This work is supported by Hong Kong RGC 14300817 (from Chinese University of Hong Kong) and Beijing Natural Science Foundation (Z180003).

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Authors and Affiliations

  1. Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Alastair King

  2. Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

    Yu Qiu

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  1. Alastair King
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  2. Yu Qiu
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Correspondence to Yu Qiu.

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King, A., Qiu, Y. Cluster exchange groupoids and framed quadratic differentials. Invent. math. 220, 479–523 (2020). https://doi.org/10.1007/s00222-019-00932-y

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